Euler's Gem: The Polyhedron Formula and the Birth of Topology

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Princeton University Press, 8 mar. 2012 - 336 páginas
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Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea.


From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.


Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.

 

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Reseña de usuario  - ztutz - LibraryThing

This book is a very well written and easy to read introduction to why topology exists. The central theme of the book is Euler's polyhedron formula, which is so simple that anyone who knows how to ... Leer reseña completa

Índice

Introduction
1
CHAPTER 1 Leonhard Euler and His Three Great Friends
10
CHAPTER 2 What Is a Polyhedron?
27
CHAPTER 3 The Five Perfect Bodies
31
CHAPTER 4 The Pythagorean Brotherhood and Platos Atomic Theory
36
CHAPTER 5 Euclid and His Elements
44
CHAPTER 6 Keplers Polyhedral Universe
51
CHAPTER 7 Eulers Gem
63
CHAPTER 17 Are They the Same or Are They Different?
173
CHAPTER 18 A Knotty Problem
186
CHAPTER 19 Combing the Hair on a Coconut
202
CHAPTER 20 When Topology Controls Geometry
219
CHAPTER 21 The Topology of Curvy Surfaces
231
CHAPTER 22 Navigating in n Dimensions
241
CHAPTER 23 Henri Poincaré and the Ascendance of Topology
253
The MillionDollar Question
265

CHAPTER 8 Platonic Solids Golf Balls Fullerenes and Geodesic Domes
75
CHAPTER 9 Scooped by Descartes?
81
CHAPTER 10 Legendre Gets It Right
87
CHAPTER 11 A Stroll through Königsberg
100
CHAPTER 12 Cauchys Flattened Polyhedra
112
CHAPTER 13 Planar Graphs Geoboards and Brussels Sprouts
119
CHAPTER 14 Its a Colorful World
130
CHAPTER 15 New Problems and New Proofs
145
CHAPTER 16 Rubber Sheets Hollow Doughnuts and Crazy Bottles
156
Acknowledgments
271
Build Your Own Polyhedra and Surfaces
273
Recommended Readings
283
Notes
287
References
295
Illustration Credits
309
Index
311
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Sobre el autor (2012)

David S. Richeson is associate professor of mathematics at Dickinson College.

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